c e r t a i n f u n c t i o n s w i t h r e s p e c t t o U D M D p r o d u c t s y s t e m s
KÁROLY NAGY
A b s t r a c t . In this paper we investigate the norm convergence of Feje'r means of functions belonging to Lipschitz classes in that case when the orthonormal system is the unitary dyadic martingale différence system (UDMD system). We give an estimation of the order of norm convergence. T h e resuit of the paper shows a sharp constrast between the corresponding known Statement which relatives to the ordinary Walsh-Paley system.
Introduction
Let N denote the set of natural numbers, P denote the set of positiv intergers, and A = {0,1}. For each m G N let [ m SJ\ j G N ) represent the binar y coefficient of m, that is,
oo
m = £ ™>{j)2j (™(i) G A, j G N).
3=0
Let (0,A) be a measure space with À(f2) = 1 and $ : = , j G N) be a sequence of A-measurable functions on Cl which are a.e. [À] bounded by 1. The product system generated by $ is the sequence of functions
: = , m G N ) defined by
oo
= n * f '
3=0
for m G N. Each is a finite product of </>n's, ipo(x) = 1 for \ipm\ <
1 for m G N, and 4>n = ip2n f °r G N.
Let $ ( i pm, m G N) be any orthonormal system on 0. The Fourier coefficients of a function / G A) are defined by
{ÎAm}-= S f'KdX ( m G N).
u
R e s e a r c h s u p p o r t e d by the H u n g á r i á n N a t i o n a l F o u n d a t i o n for Scientific Research ( O T K A ) , grant no. F 0 0 7 3 4 7
The Fourier sériés of / with respect to the system is the series oo
m=0
The partial sums of order n of the Fourier series of / are defined by n - l
Snf''= X i f,Í>m}Í>m
771 = 0
for n G P
The Cesaro means of order n of the Fourier series of an / are defined by
£ S i (nG P).
ra=1
Denote the dyadic, or 2-series, group by ( G , + ) . Thus G consists of sequences x : = j G N) with x^ = 0, or 1 and addition + is coordi- natewise, modulo 2.
Let ü = [0,1) or G. By the additive digits G N ) of an x G fi we shall mean the coordinates of x = ( x ^0) , ^1) , . . . ) if x G G and the binary coefficients of x = x ^ x ^ [0,1 ), where the finite binary expansion of x is used when x is a dyadic rational.
Let A be the Lebesgue measure when fi = [0,1) and Haar measure when Í2 = G. Denote the corresponding Lebesgue spaces by Lp(Cl) for 0 < p < oo.
By a dyadic interval of rank n in íi = [0,1) we mean an interval of the form
^ S r ^ ) w h e r e 0 < p < 2n and n G N . Given a G [0,1) and n G N , there is one and only one interval of rank n which contains a. Let it be denoted by In{.a)- By a dyadic interval of rank n centered at a G 0 = G we mean a set of the form
In(a) = {x G G: xW = a{k\k < n}.
Denote the cr-algebra generated by the intervais In(a) (a G O) by An. The intervals In(a) (a G fi) are called the atoms of An. Each element of An is a finite disjoint union of atoms.
A function / defined on Í2 is said to be ^4n-measurable if / is constant on the atoms of An. Let L(An ) denote the set of ^4n-measurable functions on fi. Each / G L(An) is integrable.
Let the Rademacher system on Q be denoted by {rn : n G N } , that is rn(x) = ( - i r( n ) ( n e N )
where { x ^ : j G N } represents the additive digits of x. Let an : fi —»C be a fonction such that each an is ^4n-measurable for any n G N . A sequence {(f)n : n G N } is said to be a UDMD system if
(f)n\=rnan (n G N)
for some ^4n-measurable fonctions an and if \<pn{x)\ = 1 (n G N, x G H) (see [3]). The simplest UDMD system is the Rademacher system. The product system generated by the UDMD system is said to be UDMD product system.
If {i/jm : m G N } is a UDMD product system then it is orthonormal on L2(Q).
The Dirichlet kernels of the product system n G N } are defined as follows Dq(X,Î) : = 0 and
n-l
O M e i î ) . j=0
for n G P- The partial sums S* f can be expressed using the Dirichlet kernels '•
( s ; / ) ( » ) = / f(t)Dï(X,t)d\(t).
Ju
The subsequence Dfn has a closed form. For every n G N
(1)
=
n a + * , - ( * ) ? * ) ) = { £ •VtluX
j=o
The Fejér kernels of the product system $ are defined by Kq (x,t) := 0 and C M e n )
n z—' m-0
for n G N . The Cesaro means of a Fourier sériés S j f can be expressed using the Fejér kernels:
K*/)(*)= f fW*(x,t)d\(t).
Jn Introduce the following notation:
{(f>k ®Jk)(x,t) := 4>k(xjfa(t) ( i , í e í ] , H N ) . For every n G N and x G Í2 (see [4]):
n—1 n—1
(2) K* = 2~nD l + 2 3~n I I í1 + ^ ®
(3) í \K%(x,t)\d\(t)<S.
Jn
Let X be a Banach space with norm || ||. The space X is called a homogeneous Banach space if P Ç X Ç where P is the set of Walsh polynomials, rxf ( y ) : = f ( y + x) and if the following three properties hold (see [1]):
( 0 ll/lk < 11/11 ( / € X ) ,
(«') r x f e x , IMII = y/11 ( x t í i j e x ) ,
and, for a given / G X there is a sequence of Walsh polynomials ( Pn, n G N ) such that
(Üt) lim ||Pn - /II = o. n—+oo
Define the modulus of continuity in X of an / G X by UX(f,6):= sup II/ — Ty/II (ő> 0).
|y|<í
For each a > 0, Lipschitz classes in X can be defined by
L i p ( a , X ) : = { / G X : ux{f,ô) = 0(6a) as 6 oo}.
Result s
T H E O R E M . Suppose / G Lip(c*,X) and a > 0. Then
W n f - f || =
' 0(n~a) 0 < a < \
as n —> oo.
PROOF. Let n G P and choose s G N such that 2s < n < 2S + 1. Let s-1
k=0
First of all we will show that
I k 2 ' / - / | | = 0(A(s)), as 5 > oo.
Since
[ K%(x,t)d\{t)
Jn „ = 1
Ju we have
<r*f(x)-f(x) = f f(t)K*(x,t)d\(t)-f(x) = [ (f(t)-f(x))K*(x,t)d\(t) JQ JU
for any / G and any x G fi. For any t G Is{x) we have
\K*(x,t)\ <2S"1. A disjoint décomposition of fi is
/s—1
« = / . ( * ) I J U Ik(x)\ik+l{x)
\k=0
for any x G fi. Let (a:) be denoted by The following inequality holds for any x G fi and any / G L i p ( a , X ) :
!<£/(*) - /(*)l < J l/W - f(x)\\K*(x,t)\d\(t) <
j \f(t)-f(x)\\K*(x,t)\d\{t)+
Is{x)
+ Ë / i / w - / ( ® ) i i ^ i ( « , o i d A ( < ) <
25-1 j \m - f(x)\d\(t)+
/.(*)
5 - 1
/c=0 /x
Lk
s —1 s —1 » s —1
x ; E
2 1 7 - 5/ i/(o - / m i n i
1+ <
fc = 0 7 = 0 rx '=0 ;=U j--
s - l , s - l
& p o — 1
2 ' -12 ' w * ( / , 2~s) + £ / | / ( í ) - / ( x ) | f ] | l f
/ C = 0 r x ' = 0
'S
S - l
k=0
/
s —1 s —1
n i i + M * W I M M < 2 -s) +
r x < = ° fc=0 Lk
From this inequality we have
s - l
1 1 4 / - /II < " * ( / , 2 "s) + E 2 2 ~ k ) = 0(A(s)) as s - oo.
k=0 for any / 6 Lip(o;,X).
We have used the following result:
T X ( = 0
Lk l^k
S - l
To prove this, let /) (1-|-(J)i(x)4>i(t)) and suppose for a moment 1=0 lïk
that
(4) ^ j f J t f i - i O M ) l2d A ( t ) < V * .
Using the Cauchy-Buniakovski inequahties we have
/ < ^ ) / / i f f ^ o i ' d A í o
^k V *
Now we will prove (4) (see [4]):
/
. s —1 s —1
r"í ' = 0 J = 0
'Jfc
c3_ie{o,i}
i I,...,fs_i6{0,l}
£s_l=£ä_l=0
«•-Y w e r ) « ' - ï w « - !1 («MA w :=
:= Z ) f x e ( x ) x ï ( * ) x 7 { t ) X i ( t ) d X ( t ) .
£ , £ G { 0 , 1 }3-2X { 0 } / x
/ X e ) x ? ( a ; ) x 7 ( í ) X c ( í A ) / 0 if £k+i = £k+i,-,£s-i = £s-i- (see [3],
r x hk
[4]). From this fact and \<f>j(x)\ = 1 ( J G N , Î G Î Î ) we get
/ |/CiK<)l2rfA(i)< i i d\(t) < ± 2 k r = r .
J hk £,£'6{0,l}ä-2x { 0 } Jx
ck + l= ik + l cs - l =fs - l *
Let P S}f and observe that
" I i - / = <£(/ -P) + ( P - f ) + tâP- P).
Using the fact {SlSJf){x) = (Smin(i,j)f){x) w e c a n show
p =
\ Ê<
5>/ -
s"/) = -
i=1 From the inequality
I I / < « * ( / , 2 - )
k * ( / - i > ) M I < / | / ( t ) - P ( t ) | | i r ; ( x , i ) | d A ( i )
< ux( f , 2 ~s) [ \K*(x,t)\dX(t)<Sux(f,2~s), Jü
that is we have and
K / - /II < lk*(/ - P)|| + Il P - /II + b i p - P\\
<9u>x(f,2-) + \WlP-P\\.
Since i
I\al P-PII = ||S* (at / - /II < | | 4 / - /II
IWn f ~ f\\ c a n estimated by A(Ő).
For 0 < a < | we have
A(s) = 0 (2~sa + 2 " t 2s (^ -a )) = 0(2~sa) = 0(n~a) as n -> oo For a > y we have
A (s) = 2-îa+2-sJ22k(*~a) = 0 (n-Q + = 0 ( ^ J j asn-^oo.
For a = \ we have
as n ^ oo.
This complétés the proof of the theorem.
I would Hke to thank Professor G. Gát for setting the prob lem and his help.
References
[1] F . , S C H I P P , W . R . W A D E , P . SIMON a n d J . P Á L , W a l s h Sériés. A n I n - troduction to dyadic Harmonie Analysis, Akadémiai Kiadó, Budapest, and Adam Hilger, Bristol and New York (1990).
[2] G. GÁT, Orthonormal System on Vilenkin Groups, Acta Math. Hun- gar. 58 (1-2) (1991), 193-198.
[3] F., SCHIPP and W. R. WADE, Norm Convergence and Summability of Fourier Sériés with Respect to Certain Product System, (To appear.) [4] G. GÁT, Vilenkin-Firoer Sériés and Limit Periodic Arithmetic Func- tions, Colleguai mathematica societatis János Bolyai 58 (1990), 315- 332.
